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We describe a variational method to solve the Holstein model for an electron coupled to dynamical, quantum phonons on an infinite lattice. The variational space can be systematically expanded to achieve high accuracy with modest computational resources (12-digit accuracy for the 1d polaron energy at intermediate coupling). We compute ground and low-lying excited state properties of the model at continuous values of the wavevector $k$ in essentially all parameter regimes. Our results for thedoi:10.1103/physrevb.60.1633 fatcat:cvunm6xcsbdpfaml6s2ldrgo5u